Gibbs
It is easy to implement this sampler:
## Julia program for Bivariate Gibbs sampler
## author: weiya <[email protected]>
## date: 2018-08-22
function bigibbs(T, rho)
x = ones(T+1)
y = ones(T+1)
for t = 1:T
x[t+1] = randn() * sqrt(1-rho^2) + rho*y[t]
y[t+1] = randn() * sqrt(1-rho^2) + rho*x[t+1]
end
return x, y
end
## example
bigibbs(100, 0.5)
Example:
We can use the following Julia program to implement this algorithm.
## Julia program for Truncated normal distribution
## author: weiya <[email protected]>
## date: 2018-08-22
# Truncated normal distribution
function rtrunormal(T, mu, sigma, mu_down)
x = ones(T)
z = ones(T+1)
# set initial value of z
z[1] = rand()
if mu < mu_down
z[1] = z[1] * exp(-0.5 * (mu - mu_down)^2 / sigma^2)
end
for t = 1:T
x[t] = rand() * (mu - mu_down + sqrt(-2*sigma^2*log(z[t]))) + mu_down
z[t+1] = rand() * exp(-(x[t] - mu)^2/(2*sigma^2))
end
return(x)
end
## example
rtrunormal(1000, 1.0, 1.0, 1.2)
In my opinion, we can illustrate this algorithm with one dimensioanl case. Suppose we want to sample from normal distribution (or uniform distribution), we can sample uniformly from the region encolsed by the coordinate axis and the density function, that is a bell shape (or a square).
Consider the normal distribution as an instance.
It is also easy to write the following Julia program.
## Julia program for Slice sampler
## author: weiya <[email protected]>
## date: 2018-08-22
function rnorm_slice(T)
x = ones(T+1)
w = ones(T+1)
for t = 1:T
w[t+1] = rand() * exp(-1.0 * x[t]^2/2)
x[t+1] = rand() * 2 * sqrt(-2*log(w[t+1])) - sqrt(-2*log(w[t+1]))
end
return x[2:end]
end
## example
rnorm_slice(100)
A special case of Completion Gibbs Sampler.
Let's illustrate the scheme with grouped counting data.
And we can obtain the following algorithm,
But it seems to be not obvious to derive the above algorithm, so I wrote some more details
Then he argues that
copies of
in each iteration is not really necessary. And briefly summary the DA algorithm:
Another example:
It seems that we do not need to derive the explicit form of
, if we can directly obtain the conditional distribution. We can use the following Julia program to sample.
## Julia program for Grouped Multinomial Data (Ex. 7.2.3)
## author: weiya <[email protected]>
## date: 2018-08-26
# call gamma function
#using SpecialFunctions
# sample from Dirichlet distributions
using Distributions
function gmulti(T, x, a, b, alpha1 = 0.5, alpha2 = 0.5, alpha3 = 0.5)
z = ones(T+1, size(x, 1)-1) # initial z satisfy `z <= x`
mu = ones(T+1)
eta = ones(T+1)
for t = 1:T
# sample from g_1(theta | y)
dir = Dirichlet([z[t, 1] + z[t, 2] + alpha1, z[t, 3] + z[t, 4] + alpha2, x[5] + alpha3])
sample = rand(dir, 1)
mu[t+1] = sample[1]
eta[t+1] = sample[2]
# sample from g_2(z | x, theta)
for i = 1:2
bi = Binomial(x[i], a[i]*mu[t+1]/(a[i]*mu[t+1]+b[i]))
z[t+1, i] = rand(bi, 1)[1]
end
for i = 3:4
bi = Binomial(x[i], a[i]*eta[t+1]/(a[i]*eta[t+1]+b[i]))
z[t+1, i] = rand(bi, 1)[1]
end
end
return mu, eta
end
# example
## data
a = [0.06, 0.14, 0.11, 0.09];
b = [0.17, 0.24, 0.19, 0.20];
x = [9, 15, 12, 7, 8];
gmulti(100, x, a, b)
Let us illuatrate this algorithm with the following example.
Last modified 4yr ago